Quantum Search Research Articles

Phase matching in quantum search algorithm

We systematically generalize the Grover algorithm in a density matrix formalism by exploiting the underlying two-dimensional subspace of the problem. Using this, we derive analytic expressions for the success probability after arbitrary iterations of the generalized Grover operator with two generic phase angles . We show for the phase matching condition with three iterations, success probability can be achieved only with knowledge about the lower bound , where λ is the ratio of marked to total number of states in the database. This result will improve the quantum search algorithm when applied to databases with unknown number of marked states in the specified regime of λ, at the cost of decreased efficiency in the smaller λ region.

Simulation of Grover Search with Polar CH3CN Molecules by Optimal Control Fields

AbstractTheoretical schemes for the implementation of the Grover search algorithm are proposed based on ultracold polar molecules in an electric field. The molecular qubits and qudit are chosen as the field‐dressed states formed by the rotational modes of CH3CN. With the help of multi‐target optimal control theory, the microwave pulses for the elementary logic operations including the one‐qubit Hadamard gates and the conditional phase gates in a dipole‐dipole system are designed and the factorized Grover algorithm demonstrated using two coupled CH3CN molecules. To reduce the accumulation of imprecision and decoherence resulting from the combination of multiple elementary gates, the optimal pulses that can achieve the two‐qubit Hadamard gate and diffusion operation in one step are designed successfully. In this way, the probabilities to find the states corresponding to the desired elements in an unsorted database are enhanced to some extent. Moreover, the optimal pulse sequence suitable for a four‐level molecular qudit, which can be used to simulate the quantum search on a single CH3CN molecule with high fidelity, is designed. These results can shed some light on the physical realization of quantum computing based on ultracold polyatomic molecules.

Invariance of success probability in Grover's quantum search under local noise with memory

We analyze the robustness of Grover's quantum search algorithm performed by a quantum register under a possibly time-correlated noise acting locally on the qubits. We model the noise as originating from an arbitrary but fixed unitary evolution, $U$, of some noisy qubits. The noise can occur with some probability in the interval between any pair of consecutive noiseless Grover evolutions. Although each run of the algorithm is a unitary process, the noise model leads to decoherence when all possible runs are considered. We derive a set of unitary $U$'s, called the 'good noises,' for which the success probability of the algorithm at any given time remains unchanged with varying the non-trivial total number ($m$) of noisy qubits in the register. The result holds irrespective of the presence of any time-correlations in the noise. We show that only when $U$ is either of the Pauli matrices $\sigma_x$ and $\sigma_z$ (which give rise to $m$-qubit bit-flip and phase-damping channels respectively in the time-correlation-less case), the algorithm's success probability stays unchanged when increasing or decreasing $m$. In contrast, when $U$ is the Pauli matrix $\sigma_y$ (giving rise to $m$-qubit bit-phase flip channel in the time-correlation-less case), the success probability at all times stays unaltered as long as the parity (even or odd) of the total number $m$ remains the same. This asymmetry between the Pauli operators stems from the inherent symmetry-breaking existing within the Grover circuit. We further show that the positions of the noisy sites are irrelevant in case of any of the Pauli noises. The results are illustrated in the cases of time-correlated and time-correlation-less noise. We find that the former case leads to a better performance of the noisy algorithm. We also discuss physical scenarios where our chosen noise model is of relevance.

Degeneration of the Grover search algorithm with depolarization in the oracle-box wires

Grover’s search algorithm and similar techniques are widely used in quantum information science. Communication lines with the so-called oracle are one of inevitable vulnerabilities of quantum search. The impact of localized dephasing and amplitude damping on Grover’s algorithm had already been discussed in recent literature. In this paper, we study the influence of depolarization in the oracle-box wires on the search process. It is shown that even low level of noise is sufficient to degenerate Grover’s algorithm. Complementarity relations between the relative entropy of coherence and the success probability in the presence of depolarization are studied as well.

Spatial search by continuous-time quantum walk on truncated simplex lattices

Quantum search via continuous-time quantum walk on truncated simplex lattices is an important example of the quantum search on the structured database. In this letter, when the search target is a set of the marked vertices, the run time of the quantum search and the critical jumping rates are derived. Quantum search with the partial information of the location of the marked vertex is also discussed. Furthermore, with the edge weights suitably adjusted, the three-stage search process can be optimized into a two-stage quantum search algorithm on the second-order truncated simplex lattice.

Circuit optimization of Grover quantum search algorithm

Circuit optimization of Grover quantum search algorithm

Noise Analysis of Grover's Quantum Search Algorithm

For searching an item in unstructured databases, Grover's quantum search algorithm offers quadratic speedup overclassical search algorithms. This paper reports 2 to 5 quantum-bit (Qubit) implementations of Grover's search algorithmusing the phase-flip method for oracle function realization without any extra ancilla qubit. A comprehensive estimation andanalysis of the theoretical and physical accuracies of the algorithm have been presented. The impact of increasing qubits onaccuracy has been computed and analyzed. The metrics delineated for comparison are the number of qubits and gates, depthof the circuit, execution time, and theoretical/physical accuracy. The results revealed a greater disparity between theoreticaland physical accuracy using a higher number of qubits perceived to be caused by noisy qubits utilized in computations. Thenovelty of the work is the investigation of variations caused by the noise in the accuracy and execution time of Grover'ssearch algorithm. The results indicate that because of noise, the accuracy of 2- and 3- qubit implementations declined by14.49% and 33.86%, whereas the execution time increased by 50% and 80%; respectively.

Quantum Search Approaches to Sampling-Based Motion Planning

Quantum Search Approaches to Sampling-Based Motion Planning

Applying Quantum Search Algorithm to Select Energy-Efficient Cluster Heads in Wireless Sensor Networks

Clustering is an effective topology control approach that evenly distributes loads across sensor nodes, enhances network scalability, and increases the lifetime in wireless sensor networks. In this paper, we propose a novel energy-efficient weighted cluster head (CH) selection approach that improves the overall performance of the network and increases energy efficiency. An optimization strategy is proposed that emphasizes adjusting the transmission range with the appropriate node density, which increases energy efficiency for intra- and inter-cluster communications to 86% and 97%, respectively. In addition, the implementation of a quantum search algorithm for choosing the CH is explained. Compared to the classical method such as EECS and HEED, the proposed quantum search algorithm has a quadratic speed-up advantage. The classical search algorithm requires N steps to find a specific element in an array of N elements, but instead of using a classical algorithm, Grover’s quantum search algorithm minimizes the complexity to O (N). In this work, an energy-efficient cluster head selection approach is illustrated through a classical weighted clustering algorithm, and its implementation is also extended through a quantum weighted search algorithm which is demonstrated by the simulation results.

Binary Classification Quantum Neural Network Model Based on Optimized Grover Algorithm.

We focus on the problem that the Grover algorithm is not suitable for the completely unknown proportion of target solutions. Considering whether the existing quantum classifier used by the current quantum neural network (QNN) to complete the classification task can solve the problem of the classical classifier, this paper proposes a binary quantum neural network classifical model based on an optimized Grover algorithm based on partial diffusion. Trial and error is adopted to extend the partial diffusion quantum search algorithm with the known proportion of target solutions to the unknown state, and to apply the characteristics of the supervised learning of the quantum neural network to binary classify the classified data. Experiments show that the proposed method can effectively retrieve quantum states with similar features. The test accuracy of BQM retrieval under the depolarization noise at the 20th period can reach 97% when the depolarization rate is 0.1. It improves the retrieval accuracy by about 4% and 10% compared with MSE and BCE in the same environment.

Robust three-qubit search algorithm in Rydberg atoms via geometric control

Rydberg atoms possess long coherence time and inherent scalability, which makes it promising to implement quantum algorithms. An exact and robust quantum search algorithm (SA) is essential to some practical applications. Here we propose a multisolution three-qubit SA by employing quantum circuit and geometric operations, in which the target states can be successfully searched with the fidelity of at least 99.8$%$ and the geometric operators guarantee robustness against systematic errors. In particular, the geometric three-qubit gate operators employed reduce the implementation time and help to resist against the detrimental influence of decoherence. Moreover, our scheme can be straightforwardly extended to multiqubit cases. We have carried out numerical simulations based on the master equation to illustrate the superiority of our scheme. We consider that our study provides an alternative method for executing quantum SAs with high success probability.

Robust quantum walk search without knowing the number of marked vertices

There has been a very large body of research on searching a marked vertex on a graph based on quantum walks, and Grover's algorithm can be regarded as a quantum walk-based search algorithm on a special graph. However, the existing quantum walk-based search algorithms suffer severely from the souffl\'{e} problem which mainly means that the success probability of finding a marked vertex could shrink dramatically even to zero when the number of search steps is greater than the right one, thus heavily reducing the robustness and practicability of the algorithm. Surprisingly, while the souffl\'{e} problem of Grover's algorithm has attracted enough attention, how to address this problem for general quantum walk-based search algorithms is missing in the literature. Here we initiate the study of overcoming the souffl\'{e} problem for quantum walk-based search algorithms by presenting a new quantum walk-based search framework that achieves robustness without sacrificing the quantum speedup. In this framework, for any adjustable parameter $\epsilon$, the quantum algorithm can find a marked vertex on an $N$-vertex {\it complete bipartite graph} with probability at least $ 1-\epsilon$, whenever the number of search steps $h$ satisfies $h \geq \ln(\frac{2}{\sqrt{\epsilon}})\sqrt{N} + 1$. Note that the algorithm need not know the exact number of marked vertices. Consequently, we obtain quantum search algorithms with stronger robustness and practicability.

Feedback‐Assisted Quantum Search by Continuous‐Time Quantum Walks

AbstractThe quantum search of a target node on a cycle graph by means of a quantum walk assisted by continuous measurement and feedback are addressed. Unlike previous spatial search approaches, where the oracle is described as a projector on the target state, a dynamical oracle implemented through a feedback Hamiltonian is considered. The idea is based on continuously monitoring the position of the quantum walker on the graph and then applying a unitary feedback operation based on the information obtained from measurement. The feedback changes the couplings between the nodes and it is optimized at each time via a numerical procedure. The stochastic trajectories describing the evolution for graphs of dimensions up to are numerically simulated, and the performance of the protocol is quantified via the average fidelity between the state of the walker and the target node. Different constraints on the control strategy are discussed. For unbounded controls, the protocol is able to quickly localize the walker on the target node. How the performance is lowered by posing an upper bound on the control couplings is then discussed. Finally, it is shown how a digital feedback protocol seems in general as efficient as the continuous bounded one.

Quantum approximate counting for Markov chains and collision counting

In this paper we show how to generalize the quantum approximate counting technique developed by Brassard, H{\o}yer and Tapp [ICALP 1998] to a more general setting: estimating the number of marked states of a Markov chain (a Markov chain can be seen as a random walk over a graph with weighted edges). This makes it possible to construct quantum approximate counting algorithms from quantum search algorithms based on the powerful ``quantum walk based search'' framework established by Magniez, Nayak, Roland and Santha [SIAM Journal on Computing 2011]. As an application, we apply this approach to the quantum element distinctness algorithm by Ambainis [SIAM Journal on Computing 2007]: for two injective functions over a set of $N$ elements, we obtain a quantum algorithm that estimates the number $m$ of collisions of the two functions within relative error $\epsilon$ by making $\tilde{O}\left(\frac{1}{\epsilon^{25/24}}\big(\frac{N}{\sqrt{m}}\big)^{2/3}\right)$ queries, which gives an improvement over the $\Theta\big(\frac{1}{\epsilon}\frac{N}{\sqrt{m}}\big)$-query classical algorithm based on random sampling when $m\ll N$.

Constructing approximately diagonal quantum gates

We study a method of producing approximately diagonal 1-qubit gates. For each positive integer, the method provides a sequence of gates that are defined iteratively from a fixed diagonal gate and an arbitrary gate. These sequences are conjectured to converge to diagonal gates doubly exponentially fast and are verified for small integers. We systemically study this conjecture and prove several important partial results. Some techniques are developed to pave the way for a final resolution of the conjecture. The sequences provided here have applications in quantum search algorithms, quantum circuit compilation, generation of leakage-free entangled gates in topological quantum computing, etc.

Quantum search on noisy intermediate-scale quantum devices

Quantum search algorithm (also known as Grover's algorithm) lays the foundation for many other quantum algorithms. Although it is very simple, its implementation is limited on noisy intermediate-scale quantum (NISQ) processors. Grover's algorithm was designed without considering the physical resources, such as depth, in the real implementations. Therefore, Grover's algorithm can be improved for NISQ devices. In this paper, we demonstrate how to implement quantum search algorithms better on NISQ devices. We present detailed benchmarks of the five-qubit quantum search algorithm on different quantum processors, including IBMQ, IonQ, and Honeywell quantum devices. We report the highest success probability of the five-qubit search algorithm compared to previous works. Our results show that designing the error-aware quantum search algorithms is possible, which can maximally harness the power of NISQ computers.

Light-shift-induced behaviors observed in momentum-space quantum walks

Over the last decade there have been many advances in studies of quantum walks (QWs) including a momentum-space QW recently realized in our spinor Bose-Einstein condensate system. This QW possessed behaviors that generally agreed with theoretical predictions; however, it also showed momentum distributions that were not adequately explained by the theory. We present a theoretical model which proves that the coherent dynamics of the spinor condensate is sufficient to explain the experimental data without invoking the presence of a thermal cloud of atoms as in the original theory. Our numerical findings are supported by an analytical prediction for the momentum distributions in the limit of zero-temperature condensates. This current model provides more complete explanations to the momentum-space QWs that can be applied to study quantum search algorithms and topological phases in Floquet-driven systems.

Hitting times of quantum and classical random walks in potential spaces

The spatial search problem is an interesting and important problem in computer science and especially the area of algorithms. The objective is a marked site to be found in a finite physical space, that can be modeled as a finite lattice or a graph. Many approaches have been developed to address this problem. Classical random walks and quantum walks are efficient models that address the spatial search problem. Quantum walks is a universal model of quantum computation and can be mapped directly to quantum circuits and consequently executed on quantum computers. Quantum walks utilized for quantum search proved to achieve significantly lower hitting times than their classical counterpart, classical random walks. The evolution space for the quantum walks as well as the classical random walks is up until now a free space. In our approach, we introduce external electrical potentials to the evolution space. We study the evolution of discrete time quantum and classical random walks in such potential spaces and the probability — hitting time on finding marked sites. We considered the differences in applied potential among neighboring sites as weights for the lattice — graph. We introduce these weights to the evolution space as an operator for the discrete time quantum walk and as coin probabilities for the classical random walk. Our results show that quantum walks again, evolve faster in the evolution space with the applied potential. Quantum walks also achieve better probability — hitting time on finding the marked site in the potential space. With the introduction of electrical potentials, quantum walks evolving in potential spaces, can lead to the development of novel quantum algorithms, where input parameters can be introduced as external potentials.

Fixed-point oblivious quantum amplitude-amplification algorithm

The quantum amplitude amplification algorithms based on Grover’s rotation operator need to perform phase flips for both the initial state and the target state. When the initial state is oblivious, the phase flips will be intractable, and we need to adopt oblivious amplitude amplification algorithm to handle. Without knowing exactly how many target items there are, oblivious amplitude amplification also suffers the “soufflé problem”, in which iterating too little “undercooks” the state and too much “overcooks” the state, both resulting in a mostly non-target final state. In this work, we present a fixed-point oblivious quantum amplitude-amplification (FOQA) algorithm by introducing damping based on methods proposed by A. Mizel. Moreover, we construct the quantum circuit to implement our algorithm under the framework of duality quantum computing. Our algorithm can avoid the “soufflé problem”, meanwhile keep the square speedup of quantum search, serving as a subroutine to improve the performance of quantum algorithms containing oblivious amplitude amplification procedure.

Metasurface-Based Quantum Searcher on a Silicon-On-Insulator Chip.

Optical analog computing has natural advantages of parallel computation, high speed and low energy consumption over traditional digital computing. To date, research in the field of on-chip optical analog computing has mainly focused on classical mathematical operations. Despite the advantages of quantum computing, on-chip quantum analog devices based on metasurfaces have not been demonstrated so far. In this work, based on a silicon-on-insulator (SOI) platform, we illustrated an on-chip quantum searcher with a characteristic size of 60 × 20 μm2. We applied classical waves to simulate the quantum search algorithm based on the superposition principle and interference effect, while combining it with an on-chip metasurface to realize modulation capability. The marked items are found when the incident waves are focused on the marked positions, which is precisely the same as the efficiency of the quantum search algorithm. The proposed on-chip quantum searcher facilitates the miniaturization and integration of wave-based signal processing systems.